Cauchy problems for fifth-order KdV equations in weighted Sobolev spaces

In this work we study the initial-value problem for the fifth-order Korteweg-de Vries equation $$ \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0, \quad x,t\in \mathbb{R}, \; k=1,2, $$ in weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(\langle x \rangle^{2r}dx)$. We prove local and global results. For the case $k=2$ we point out the relationship between decay and regularity of solutions of the initial-value problem